Abstract

For a rotation invariant domain Ξ©, we consider 𝐴2(Ξ©,πœ‡) the Bergman space and we investigate some properties of the rank one projection 𝐴(𝑧)∢=βŸ¨β‹…,π‘˜π‘§βŸ©π‘˜π‘§. We prove that the trace of all the strong derivatives of A(z) is zero. We also focus on the generalized Fock space 𝐴2(πœ‡π‘š), where πœ‡π‘š is the measure with weight π‘’βˆ’|𝑧|π‘š, π‘š>0, with respect to the Lebesgue measure on ℂ𝑛 and establish estimations of derivatives of the Berezin transform of a bounded operator T on 𝐴2(πœ‡π‘š).

1. Introduction and Statement of the Main Results

We consider a rotation invariant open set Ξ© in ℂ𝑛 and πœ‡ a positive rotation invariant measure on Ξ©; we suppose that πœ‡ has moments of every order. Let 𝐿2(Ξ©,πœ‡) be the Hilbert space of square integrable complex-valued functions on Ξ© and π’œ2(Ξ©,πœ‡) its subspace consisting of holomorphic elements. We assume that for each compact set πΎβŠ‚Ξ© there exists 𝐢=𝐢(𝐾) such that for all π‘“βˆˆπ’œ2(Ξ©,πœ‡)supπ‘§βˆˆπΎ||||𝑓(𝑧)≀𝐢‖𝑓‖𝐿2(Ξ©,πœ‡).(1.1) It is known that π’œ2(Ξ©,πœ‡) is a closed space of 𝐿2(Ξ©,πœ‡) and possesses a reproducing kernel 𝐾(πœ”,𝑧)∢=𝐾𝑧(πœ”): we haveξ€œπ‘“(𝑧)=Ω𝑓(πœ”)𝐾(𝑧,πœ”)π‘‘πœ‡(πœ”)=βŸ¨π‘“,πΎπ‘§βŸ©,(1.2) for all π‘“βˆˆπ’œ2(Ξ©,πœ‡) and π‘§βˆˆΞ©.

For a bounded linear operator 𝑇 on π’œ2(Ξ©,πœ‡), the Berezin transform of 𝑇 is the function 𝑇 defined on Ξ© by𝑇(𝑧)∢=βŸ¨π‘‡π‘˜π‘§,π‘˜π‘§βŸ©,(1.3) where π‘˜π‘§ is the normalized reproducing kernelπ‘˜π‘§(πœ”)∢=𝐾(πœ”,𝑧)𝐾(𝑧,𝑧)1/2.(1.4) The case of the Fock space, where Ξ©=ℂ𝑛 and πœ‡2 is the Gaussian measure, was considered by Coburn, Englis, and Zhang. Coburn [1] has shown that 𝑇 is a Lipschitz function. Namely, for π‘₯,π‘¦βˆˆβ„‚π‘›,||𝑇𝑇||||||,(π‘₯)βˆ’(𝑦)≀2‖𝑇‖π‘₯βˆ’π‘¦(1.5) the constant 2 being sharp (see [2]).

Englis and Zhang [3] have shown that 𝑇 has bounded derivatives of all orders. Namely, for any multi-indices 𝛽, 𝛾, there exists a constant 𝑐𝛽,𝛾, depending on 𝛽, 𝛾, and 𝑛 only, such thatβ€–β€–πœ•π›½πœ•π›Ύξ‚π‘‡β€–β€–βˆžβ‰€π‘π›½,𝛾‖𝑇‖,(1.6) where πœ•π›½ will stand forπœ•|𝛽|πœ•π‘§π›½11β‹―πœ•π‘§π›½π‘›π‘›πœ•,whereπœ•π‘§π‘—=12ξ‚΅πœ•πœ•π‘₯π‘—πœ•βˆ’π‘–πœ•π‘¦π‘—ξ‚Άfor𝑧𝑗=π‘₯𝑗+𝑖𝑦𝑗(1.7) and similarly for πœ•. Recently, the author extended Coburn’s result to weighted Fock spaces, corresponding to Ξ©=ℂ𝑛 endowed with the measureπ‘‘πœ‡π‘š(𝑧)=π‘’βˆ’|𝑧|π‘šπ‘‘π‘‰(𝑧),(1.8) where π‘š is a positive parameter and 𝑑𝑉 is the normalized Lebesgue measure on ℂ𝑛, such that the volume of the unit ball in ℂ𝑛 is equal to 2𝑛. We have showed that 𝑇 satisfies a local Lipschitz condition. There exist positive constants 𝐴, 𝐢, depending on 𝑛 and π‘š only, such that, for any π‘Žβˆˆβ„‚π‘› with |π‘Ž|>𝐴, there is a neighbourhood π‘ˆ of π‘Ž that satisfies||𝑇𝑇||(π‘Ž)βˆ’(π‘₯)≀𝐢‖𝑇‖|π‘Ž|(π‘š/2)βˆ’1|π‘₯βˆ’π‘Ž|,βˆ€π‘₯inπ‘ˆ.(1.9) In this paper we will investigate some properties of the derivatives of the Berezin transform on π’œ2(Ξ©,πœ‡). For 𝑧 in Ξ©, if 𝐴(𝑧) is the rank one projection𝐴(𝑧)∢=βŸ¨β‹…,π‘˜π‘§βŸ©π‘˜π‘§,(1.10) we have𝑇(𝑧)=tr(𝑇𝐴(𝑧)).(1.11) We first fix some notations. Let ℕ𝑛0 denote the set of all 𝑛-tuples with components in the set β„•0 of all nonnegative integers. If 𝛽=(𝛽1,…,𝛽𝑛)βˆˆβ„•π‘›0, we let |𝛽|∢=𝛽1+β‹―+𝛽𝑛 denote the length of 𝛽 and 𝛽! stands for βˆπ‘›π‘—=1𝛽𝑗!. If 𝛾=(𝛾1,…,𝛾𝑛)βˆˆβ„•π‘›0 satisfies 𝛾𝑗≀𝛽𝑗 for all 𝑗=1,…,𝑛, then we write 𝛾≀𝛽. Our first main result is about the strong derivatives of 𝐴(𝑧).

Theorem 1.1. Let Ξ© be a rotation invariant open set in ℂ𝑛 and πœ‡ a rotation invariant positive measure on Ξ© that satisfies (1.1) and has moments of every order. Moreover one assumes that, for any multi-index 𝛼 and any compact set 𝐾 of Ξ©, there exists 𝐺𝛼,𝐾∈𝐿2(Ξ©,πœ‡) such that ||π‘€π›Όπœ•π›Ό||𝐾(𝑧,𝑀)≀𝐺𝛼,𝐾(𝑀),βˆ€π‘§βˆˆπΎ.(1.12) Then for all 𝛽, 𝛾 multi-indices, the operators πœ•π›½πœ•π›Ύπ΄(𝑧) and πœ•π›½πœ•π›Ύπ΄(𝑧) are adjoint to each other; their rank is smaller or equal to the infimum of #{π›Ώβˆˆβ„•π‘›0,𝛿≀𝛾} and #{π›Ώβˆˆβ„•π‘›0,𝛿≀𝛽}.
Moreover, if at least 𝛽 or 𝛾 is different from 0, one has ξ‚ƒπœ•trπ›½πœ•π›Ύξ‚„π΄(𝑧)=0.(1.13)

Our second main result generalizes the estimates of Englis and Zhang for the strong derivatives of the Berezin transform on weighted Fock spaces.

Theorem 1.2. For a bounded linear operator 𝑇 on π’œ2(ℂ𝑛,πœ‡π‘š), the Berezin transform 𝑇 has derivatives of all orders. In addition to any multi-indices 𝛽, 𝛾, there exist positive constants 𝑐𝛽,𝛾 and 𝐴, depending on 𝛽, 𝛾, and 𝑛 only such that |||πœ•π›½πœ•π›Ύξ‚|||𝑇(𝑧)≀𝑐𝛽,𝛾|𝑧|((π‘š/2)βˆ’1)(|𝛽+𝛾|)for|𝑧|>𝐴.(1.14)

2. Preliminaries

We recall some properties of the Bergman kernel 𝐾, when πœ‡ is a positive rotation invariant measure on Ξ©. The kernel 𝐾 is given by𝐾(𝑧,πœ”)=π›½βˆˆβ„•π‘›01π‘π›½π‘§π›½πœ”π›½,(2.1) for 𝑧, πœ” in Ξ©, with the usual convention that, for 𝑧=(𝑧1,…,𝑧𝑛) and 𝛽=(𝛽1,…,𝛽𝑛), 𝑧𝛽 stands for 𝑧𝛽11⋯𝑧𝛽𝑛𝑛 and whereπ‘π›½ξ€œβˆΆ=Ξ©π‘₯𝛽π‘₯π›½π‘‘πœ‡(π‘₯).(2.2) By Lemma  2.1 in [4], we know that𝑐𝛽=(π‘›βˆ’1)!𝛽!π‘š|𝛽|ξ€·||𝛽||ξ€Έ!π‘›βˆ’1+,whereπ‘šπ‘˜=ξ€œΞ©|𝑧|2π‘˜π‘‘πœ‡(𝑧)forπ‘˜βˆˆβ„•0.(2.3) Thus we can write𝐾(𝑧,πœ”)=𝐹(βŸ¨π‘§,πœ”βŸ©),(2.4) where1𝐹(𝑑)∢=(ξ“π‘›βˆ’1)!π‘‘βˆˆβ„•0(π‘›βˆ’1+𝑑)!π‘šπ‘‘π‘‘π‘‘(2.5) is a holomorphic function of one complex variable.

For a bounded operator 𝑇 on π’œ2(Ξ©,πœ‡), the Berezin transform can be written in the form𝑇(𝑧)=tr(𝑇𝐴(𝑧)),forπ‘§βˆˆΞ©,(2.6) where 𝐴(𝑧) is the rank one projection 𝐴(𝑧)=βŸ¨β‹…,π‘˜π‘§βŸ©π‘˜π‘§.

Recall [3] that a mapping β„Ž from a domain in ℂ𝑛 into a Banach space possesses a strong holomorphic derivative πœ•β„Ž/πœ•π‘§1(𝑧) at a point 𝑧=(𝑧1,…,𝑧𝑛)βˆˆβ„‚π‘› iflim𝑑→0β€–β€–β€–β€–β„Žξ€·π‘§1+𝑑,𝑧2,…,π‘§π‘›ξ€Έξ€·π‘§βˆ’β„Ž1,…,π‘§π‘›ξ€Έπ‘‘βˆ’πœ•β„Žπœ•π‘§1𝑧1,…,𝑧𝑛‖‖‖‖=0;(2.7) similarly one can define the antiholomorphic derivative πœ•. Englis and Zhang showed that the mapping 𝑧↦𝐴(𝑧) has strong derivatives.

Lemma 2.1. Let Ξ© be a domain in ℂ𝑛, πœ‡ a measure on Ξ© that satisfies (1.1). Then the function π‘§β†¦π‘˜π‘§ has strong derivatives of all orders.

Lemma 2.2. Under the same hypothesis, the trace-class-operator-valued function π‘§βŸΌβŸ¨β‹…,π‘˜π‘§βŸ©π‘˜π‘§(2.8) from Ξ© into the space of trace-class operators on π’œ2(Ξ©,πœ‡) has strong derivatives of all orders.

The mapping 𝑧↦𝐴(𝑧) and its derivatives can be expressed in terms of the function 𝐹. It is easy tosee that, for 𝑧, 𝑒 in Ξ©,𝐴(𝑧)⋅𝑓(𝑒)=𝑓,𝐹(βŸ¨β‹…,π‘§βŸ©)𝐹(βŸ¨π‘’,π‘§βŸ©)𝐹(βŸ¨π‘§,π‘§βŸ©).(2.9)

3. Proof of Theorem 1.1

We write, for π‘“βˆˆπ’œ2(Ξ©,πœ‡) and 𝑧, 𝑒 in Ξ©,ξ‚Έξ€œπ΄(𝑧)⋅𝑓(𝑒)=Ω𝑓(πœ”)𝐹(βŸ¨π‘§,πœ”βŸ©)π‘‘πœ‡(πœ”)𝐹(βŸ¨π‘’,π‘§βŸ©)𝐹(βŸ¨π‘§,π‘§βŸ©).(3.1) Since it is possible to differentiate under the integral sign at any order with respect to 𝑧, for 𝛽 in ℕ𝑛0, we haveξ€·πœ•π›½ξ€Έξ€œπ΄(𝑧)⋅𝑓(𝑒)=Ω𝑓(πœ”)πœ•π›½ξ‚ΈπΉ(βŸ¨π‘§,πœ”βŸ©)𝐹(βŸ¨π‘§,π‘§βŸ©)π‘‘πœ‡(πœ”)𝐹(βŸ¨π‘’,π‘§βŸ©).(3.2) The Leibnitz rule leads toξ€·πœ•π›½ξ€Έξ“π΄(𝑧)⋅𝑓(𝑒)=πœˆβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ ξ€œΞ©π‘“(πœ”)πœ”πœˆπΉ(|𝜈|)(βŸ¨π‘§,πœ”βŸ©)π‘§π›½βˆ’πœˆξ‚€1𝐹(|π›½βˆ’πœˆ|)(βŸ¨π‘§,π‘§βŸ©)π‘‘πœ‡(πœ”)𝐹(βŸ¨π‘’,π‘§βŸ©).(3.3) Setting ̇𝐾𝑧(𝛽)(πœ”)=𝐹(βŸ¨π‘§,π‘§βŸ©)1/2βˆ‘πœˆβŠ‚π›½ξ€·π›½πœˆξ€Έπœ”πœˆπ‘§π›½βˆ’πœˆ(1/𝐹)(|π›½βˆ’πœˆ|)(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜈|)(βŸ¨πœ”,π‘§βŸ©), we getπœ•π›½ξ‚¬Μ‡πΎπ΄(𝑧)⋅𝑓=𝑓,𝑧(𝛽)ξ‚­π‘˜π‘§(3.4) and thenπœ•π›Ύπœ•π›½π΄ξ“(𝑧)⋅𝑓=𝛿≀𝛾𝑓,𝐿𝑧,𝛿𝑀𝑧,𝛿,(3.5) where 𝐿𝑧,𝛿(πœ”)=(πœ•π›Ώ/πœ•π‘§π›Ώ)(𝐹(βŸ¨π‘§,π‘§βŸ©)βˆ’1/2̇𝐾𝑧(𝛽)(πœ”)) and 𝑀𝑧,𝛿(𝑒)=π›Ύπ›Ώξ€Έπ‘’π›Ύβˆ’π›ΏπΉ(|π›Ύβˆ’π›Ώ|)(βŸ¨π‘’,π‘§βŸ©). Notice that the rank of the operator πœ•π›Ύπœ•π›½π΄(𝑧) is smaller than or equal to #{π›Ώβˆˆβ„•π‘›0,𝛿≀𝛾}.

Setting now π‘†βˆΆ=πœ•π›Ύπœ•π›½π΄(𝑧), πΏπ›ΏβˆΆ=𝐿𝑧,𝛿, and π‘€π›ΏβˆΆ=𝑀𝑧,𝛿 for a fixed element 𝑧 in Ξ©, we obtain, for 𝑓,𝑔 in π’œ2(Ξ©,πœ‡),ξ„”ξ“βŸ¨π‘†π‘“,π‘”βŸ©=π›Ώβ‰€π›ΎβŸ¨π‘“,πΏπ›ΏβŸ©π‘€π›Ώξ„•=,𝑔𝑓,π›Ώβ‰€π›ΎβŸ¨π‘”,π‘€π›ΏβŸ©πΏπ›Ώξ„•.(3.6) Consequently,π‘†βˆ—ξ“π‘”=π›Ώβ‰€π›ΎβŸ¨π‘”,π‘€π›ΏβŸ©πΏπ›Ώ,(3.7) that is,ξ€·π‘†βˆ—π‘”ξ€Έξ“(𝑒)=π›Ώβ‰€π›ΎβŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ€œΞ©π‘”(πœ”)πœ”π›Ύβˆ’π›ΏπΉ(|π›Ύβˆ’π›Ώ|)πœ•(βŸ¨π‘§,πœ”βŸ©)π‘‘πœ‡(πœ”)π›Ώπœ•π‘§π›Ώξƒ©1βˆšΜ‡πΎπΉ(βŸ¨π‘§,π‘§βŸ©)𝑧(𝛽)ξƒͺ=πœ•(𝑒)π›Ύπœ•π‘§π›Ύξƒ¬ξ€œΞ©ξƒ©1𝑔(πœ”)𝐹(βŸ¨π‘§,πœ”βŸ©)π‘‘πœ‡(πœ”)βˆšΜ‡πΎπΉ(βŸ¨π‘§,π‘§βŸ©)𝑧(𝛽)=πœ•(𝑒)ξƒͺξƒ­π›Ύπœ•π‘§π›Ύξ‚ƒβŸ¨π‘”,π‘˜π‘§βŸ©Μ‡πΎπ‘§(𝛽)ξ‚„=πœ•(𝑒)π›Ύπœ•π‘§π›Ύπœ•π›½πœ•π‘§π›½ξ€ΊβŸ¨π‘”,π‘˜π‘§βŸ©π‘˜π‘§ξ€»=ξ‚€πœ•(𝑒)π›Ύπœ•π›½π΄ξ‚(𝑧)⋅𝑔(𝑒).(3.8) It follows that the rank of the operator πœ•π›Ύπœ•π›½π΄(𝑧) is also smaller than or equal to #{π›Ώβˆˆβ„•π‘›0,𝛿≀𝛽}. Thusξ‚€πœ•π›Ύπœ•π›½ξ‚π΄(𝑧)βˆ—=πœ•π›Ύπœ•π›½π΄(𝑧),forπ‘§βˆˆΞ©.(3.9) Now let 𝛽 and 𝛾 be multi-indices. Due to Lemma 2.2 and the continuity of the linear form Tr(𝑋), we haveξ‚€πœ•Trπ›Ύπœ•π›½ξ‚π΄(𝑧)=πœ•π›Ύπœ•π›½Tr𝐴(𝑧)=πœ•π›Ύπœ•π›½1=𝛿𝛽0𝛿𝛾0.(3.10)

4. Proof of Theorem 1.2

When Ξ©=ℂ𝑛 and π‘‘πœ‡π‘š(𝑧)=π‘’βˆ’|𝑧|π‘šπ‘‘π‘£(𝑧), for brevity we set π’œ2(πœ‡π‘š) for π’œ2(ℂ𝑛,πœ‡π‘š). The Bergman kernel πΎπ‘š of π’œ2(πœ‡π‘š) can be expressed in terms of the Mittag-Leffler function (see [5]). Putting 𝛼=2/π‘š, we haveπΎπ‘š(𝑧,πœ”)=𝐹(βŸ¨π‘§,πœ”βŸ©),where𝐹(𝑑)=𝐢𝐸(π‘›βˆ’1)𝛼,π›Όπ‘š(𝑑),𝐢=,(π‘›βˆ’1)!(4.1)𝐸(π‘›βˆ’1)𝛼,𝛼 being the π‘›βˆ’1-th derivative of the Mittag-Leffler function 𝐸𝛼,𝛼, entire on β„‚ and defined by𝐸𝛼,𝛼(𝑑)∢=+βˆžξ“π‘‘=0𝑑𝑑Γ(𝛼𝑑+𝛼),forπ‘‘βˆˆβ„‚.(4.2) In what follows, we will use some asymptotic properties of this function (see [6]) near the positive real axis.

Lemma 4.1. Let 𝑝 a nonnegative integer. There exists a constant πœ–>0, such that, for any complex number 𝑧 with |arg𝑧|<πœ–, one has 𝐹(𝑝)(𝑧)=𝐹0(𝑝)(𝑧)+πœ–π‘(𝑧),(4.3) where 𝐹0(𝑧)=𝐢𝑒𝑍0(𝑧)𝑍0(𝑧)βˆ’π‘›π›Όπ‘ƒπ‘›ξ€·π‘0ξ€Έ(𝑧)(4.4) and 𝑍𝑠(𝑧)=|𝑧|1/𝛼exp[(𝑖/𝛼)(argz+2πœ‹π‘ )], for 𝑠 integer.
The polynomials (𝑃𝑝)π‘βˆˆβ„• are defined by (𝑑𝑝/𝑑𝑒𝑝)[𝑒𝑒1/𝛼]=(𝑒𝑒1/𝛼/𝑒𝑝)𝑃𝑝(𝑒1/𝛼), 𝑃𝑝 is of degree 𝑝, and when |𝑧|β†’+∞, πœ–π‘(π‘Ÿ)(𝑧)=π‘œ(𝐹0(π‘ž)(𝑍0)) for any nonnegative integers π‘Ÿ and π‘ž.

We also need asymptotic estimates for some auxiliaries functions.

Lemma 4.2. Let 𝑝 and 𝑑 be nonnegative integers. When 𝑑 is real and tends to +∞, ξ‚€1𝐹(𝑑)ξ‚΅1(𝑑)∼𝐹0ξ‚Ά(𝑑)𝐹(𝑑),(𝑝)𝐹(𝑑)𝐹(𝑑)∼0(𝑝)𝐹0ξƒͺ(𝑑)(𝑑).(4.5)

For 𝑇 a bounded linear operator on π’œ2(πœ‡π‘š), the Berezin transform is given by (see [3])𝑇(𝑧)=tr(𝑇𝐴(𝑧)).(4.6) Let 𝛽 and 𝛾 be some multi-indices. By differentiation, Lemma 2.2 givesπœ•π›½πœ•π›Ύξ‚ξ‚€π‘‡(𝑧)=trπ‘‡πœ•π›½πœ•π›Ύξ‚π΄(𝑧).(4.7) Due to the continuity of the bilinear form (𝑋,π‘Œ)↦tr(π‘‹π‘Œ), we have||||tr(π‘‹π‘Œ)β‰€β€–π‘‹β€–β€–π‘Œβ€–tr,(4.8) for all bounded operators 𝑋 and trace class operators π‘Œ. Therefore, we obtain|||πœ•π›½πœ•π›Ύξ‚|||β€–β€–πœ•π‘‡(𝑧)β‰€β€–π‘‡β€–π›½πœ•π›Ύβ€–β€–π΄(𝑧)tr.(4.9) Like in Section 3, we set π‘†βˆΆ=πœ•π›½πœ•π›Ύπ΄(𝑧). Next we shall estimate tr((π‘†π‘†βˆ—)1/2).

Fix 𝑓 in π’œ2(πœ‡π‘š) and 𝑧 in ℂ𝑛. We recall that𝑆𝑓=𝛿≀𝛾𝑓,𝐿𝑧,𝛿𝑀𝑧,𝛿,(4.10) where 𝐿𝑧,𝛿(πœ”)=(πœ•π›Ώ/πœ•π‘§π›Ώ)(𝐹(βŸ¨π‘§,π‘§βŸ©)βˆ’1/2̇𝐾𝑧(𝛽)(πœ”)) and 𝑀𝑧,𝛿(𝑒)=π›Ύπ›Ώξ€Έπ‘’π›Ύβˆ’π›ΏπΉ(|π›Ύβˆ’π›Ώ|)(βŸ¨π‘’,π‘§βŸ©). Thenπ‘†π‘†βˆ—ξ“π‘“=π›Ώβ‰€π›Ύξƒ¬ξ“π›Ώξ…žβ‰€π›Ύξ«π‘“,𝑀𝑧,𝛿′𝐿𝑧,𝛿′,𝐿𝑧,𝛿𝑀𝑧,𝛿.(4.11) To estimate the eigenvalues of the finite rank positive operator π‘†π‘†βˆ—, we compute its trace:ξ€·trπ‘†π‘†βˆ—ξ€Έ=ξ“π›Ώβ‰€π›Ύπ‘Žπ›Ώ,withπ‘Žπ›Ώ=ξ“π›Ώξ…žβ‰€π›Ύξ«π‘€π‘§,𝛿,𝑀𝑧,𝛿′𝐿𝑧,𝛿′,𝐿𝑧,𝛿.(4.12) With the operator being of finite rank, we observe that Tr(π‘†π‘†βˆ—)=Tr(π‘†βˆ—π‘†). We now compute and then estimate each diagonal term π‘Žπ›Ώ. Fix some multi-indices 𝛿 and 𝛿′ such that 𝛿≀𝛾 and 𝛿′≀𝛾. To simplify the notation we set 𝐿𝛿 and 𝑀𝛿 instead of 𝐿𝑧,𝛿 and 𝑀𝑧,𝛿. We obtain𝑀𝛿,𝑀𝛿′=𝐢2βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ ξ€œβ„‚π‘›πœ”π›Ύβˆ’π›ΏπΉ(|π›Ύβˆ’π›Ώ|)(βŸ¨πœ”,π‘§βŸ©)πœ”π›Ύβˆ’π›Ώβ€²πΉ(|π›Ύβˆ’π›Ώβ€²|)(βŸ¨π‘§,πœ”βŸ©)π‘‘πœ‡(πœ”)=𝐢2βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ 1πΆπœ•π›Ύβˆ’π›Ώπœ•π‘§π›Ύβˆ’π›Ώπœ•π›Ύβˆ’π›Ώβ€²πœ•π‘§π›Ύβˆ’π›Ώβ€²βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›ΎβŽžβŽŸβŽŸβŽ πœ•πΉ(βŸ¨π‘§,π‘§βŸ©)=πΆπ›Ώβ€²π›Ύβˆ’π›Ώπœ•π‘§π›Ύβˆ’π›Ώξ‚ƒπ‘§π›Ύβˆ’π›Ώβ€²πΉ(|π›Ύβˆ’π›Ώβ€²|)ξ‚„βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›ΎβŽžβŽŸβŽŸβŽ βŽ§βŽͺ⎨βŽͺβŽ©ξ“(βŸ¨π‘§,π‘§βŸ©)=πΆπ›Ώβ€²πœ…β‰€π›Ύβˆ’π›Ώ,πœ…β‰€π›Ύβˆ’π›Ώξ…žβŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ Γ—π›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώβ€²πœ…!π‘§π›Ύβˆ’π›Ώβ€²βˆ’πœ…π‘§π›Ύβˆ’π›Ώβˆ’πœ…πΉ(|π›Ύβˆ’π›Ώβ€²+π›Ύβˆ’π›Ώβˆ’πœ…|)⎫βŽͺ⎬βŽͺ⎭.(βŸ¨π‘§,π‘§βŸ©)(4.13) For the second term, sinceπΏπ›Ώπœ•(πœ”)=π›Ώπœ•π‘§π›ΏβŽ§βŽͺ⎨βŽͺβŽ©ξ“πœˆβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ πœ”π›½βˆ’πœˆπΉ(|π›½βˆ’πœˆ|)(βŸ¨πœ”,π‘§βŸ©)π‘§πœˆξ‚€1𝐹(|𝜈|)⎫βŽͺ⎬βŽͺ⎭=(βŸ¨π‘§,π‘§βŸ©)πœˆβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ πœ”π›½βˆ’πœˆπΉ(|π›½βˆ’πœˆ|)(πœ•βŸ¨πœ”,π‘§βŸ©)π›Ώπœ•π‘§π›Ώξ‚Έπ‘§πœˆξ‚€1𝐹(|𝜈|)(ξ‚Ή,βŸ¨π‘§,π‘§βŸ©)(4.14) we get𝐿𝛿′,𝐿𝛿=ξ“πœˆ,πœˆξ…žβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βŽžβŽŸβŽŸβŽ πœ•πœˆβ€²π›Ώξ…žπœ•π‘§π›Ώξ…žξ‚Έπ‘§πœˆβ€²ξ‚€1𝐹(|πœˆβ€²|)ξ‚Ήπœ•(βŸ¨π‘§,π‘§βŸ©)π›Ώπœ•π‘§π›ΏΓ—ξ‚Έπ‘§πœˆξ‚€1𝐹(|𝜈|)ξ‚Ήπœ•(βŸ¨π‘§,π‘§βŸ©)π›½βˆ’πœˆξ…žπœ•π‘§π›½βˆ’πœˆξ…žπœ•π›½βˆ’πœˆπœ•π‘§π›½βˆ’πœˆξ‚ΈπΉ(βŸ¨π‘§,π‘§βŸ©)𝐢,𝐿(4.15)𝛿′,𝐿𝛿=1πΆξ“πœˆ,πœˆξ…žβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βŽžβŽŸβŽŸβŽ πœ•πœˆβ€²π›Ώξ…žπœ•π‘§π›Ώξ…žξ‚Έπ‘§π›½βˆ’πœˆβ€²ξ‚€1𝐹(|π›½βˆ’πœˆβ€²|)ξ‚Ήπœ•(βŸ¨π‘§,π‘§βŸ©)π›Ώπœ•π‘§π›ΏΓ—ξ‚Έπ‘§π›½βˆ’πœˆξ‚€1𝐹(|π›½βˆ’πœˆ|)ξ‚Ήπœ•(βŸ¨π‘§,π‘§βŸ©)πœˆξ…žπœ•π‘§πœˆξ…žξ€Ίπ‘§πœˆπΉ(|𝜈|)ξ€».(βŸ¨π‘§,π‘§βŸ©)(4.16) By the Leibnitz rule, we see thatπœ•π›Ώπœ•π‘§π›Ώξ‚Έπ‘§π›½βˆ’πœˆξ‚€1𝐹(|π›½βˆ’πœˆ|)ξ‚Ή=(βŸ¨π‘§,π‘§βŸ©)πœ†β‰€π›Ώ,πœ†β‰€π›½βˆ’πœˆβŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ†βŽžβŽŸβŽŸβŽ π›½βˆ’πœˆπœ†!π‘§π›½βˆ’πœˆβˆ’πœ†π‘§π›Ώβˆ’πœ†ξ‚€1𝐹(|π›½βˆ’πœˆ+π›Ώβˆ’πœ†|)πœ•(βŸ¨π‘§,π‘§βŸ©),π›Ώξ…žπœ•π‘§π›Ώξ…žξ‚Έπ‘§π›½βˆ’πœˆβ€²ξ‚€1𝐹(|π›½βˆ’πœˆβ€²|)(ξ‚Ή=ξ“βŸ¨π‘§,π‘§βŸ©)πœ†ξ…žβ‰€π›Ώξ…ž,πœ†ξ…žβ‰€π›½βˆ’πœˆξ…žβŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ†π›Ώβ€²πœ†β€²π›½βˆ’πœˆβ€²πœ†β€²ξ…ž!π‘§π›½βˆ’πœˆβ€²βˆ’πœ†β€²π‘§π›Ώβ€²βˆ’πœ†β€²Γ—ξ‚€1𝐹(|π›½βˆ’πœˆβ€²+π›Ώβ€²βˆ’πœ†β€²|)πœ•(βŸ¨π‘§,π‘§βŸ©),πœˆξ…žπœ•π‘§πœˆξ…žξ€Ίπ‘§πœˆπΉ(|𝜈|)(ξ€»=ξ“βŸ¨π‘§,π‘§βŸ©)πœŽβ‰€πœˆ,πœŽβ‰€πœˆβ€²βŽ›βŽœβŽœβŽπœˆξ…žπœŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœˆπœŽβŽžβŽŸβŽŸβŽ πœŽ!π‘§πœˆβˆ’πœŽπ‘§πœˆβ€²βˆ’πœŽπΉ(|𝜈+πœˆβ€²βˆ’πœŽ|)(βŸ¨π‘§,π‘§βŸ©).(4.17) Then 𝑀𝛿,𝑀𝛿′𝐿𝛿′,𝐿𝛿=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›ΎβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²πœˆβ‰€π›½,πœˆξ…žβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βŽžβŽŸβŽŸβŽ ξ“πœˆβ€²πœ†β‰€π›Ώ,π›½βˆ’πœˆπœ†ξ…žβ‰€π›Ώξ…ž,π›½βˆ’πœˆξ…žπœŽβ‰€πœˆ,πœˆξ…žπœ…β‰€π›Ύβˆ’π›Ώ,π›Ύβˆ’π›Ώξ…žβŽ›βŽœβŽœβŽπœŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœˆπœŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ Γ—βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ Γ—βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ πœˆβ€²πœŽ!π›Ώβ€²πœ†β€²π›½βˆ’πœˆβ€²πœ†β€²πœ†β€²!π›½βˆ’πœˆπœ†!π›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώβ€²πœ…!𝑧𝛽+π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœŽβˆ’πœ…π‘§π›½+π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœŽβˆ’πœ…Γ—πΉ(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜈+πœˆβ€²βˆ’πœŽ|)ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆβ€²+π›Ώβ€²βˆ’πœ†β€²|)ξ‚€1Γ—(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆ+π›Ώβˆ’πœ†|)(βŸ¨π‘§,π‘§βŸ©).(4.18) Thus π‘Žπ›Ώ=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ώξ…žβ‰€π›ΎβŽ›βŽœβŽœβŽπ›ΎβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²πœˆβ‰€π›½πœˆξ…žβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βŽžβŽŸβŽŸβŽ ξ“πœˆβ€²πœ†β‰€π›Ώ,π›½βˆ’πœˆπœ†ξ…žβ‰€π›Ώξ…ž,π›½βˆ’πœˆξ…žπ›½βˆ’πœˆ,π›½βˆ’πœˆξ…žβ‰€πœŒπœ…β‰€π›Ύβˆ’π›Ώ,π›Ύβˆ’π›Ώξ…žβŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœˆβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ πœˆβ€²π›½βˆ’πœŒπ›½βˆ’πœŒ(π›½βˆ’πœŒ)!π›Ώβ€²πœ†β€²π›½βˆ’πœˆβ€²πœ†β€²Γ—πœ†β€²!π›½βˆ’πœˆπœ†!π›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώβ€²πœ…!π‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒπ‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒΓ—πΉ(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜈+πœˆβ€²βˆ’π›½+𝜌|)(ξ‚€1βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆβ€²+π›Ώβ€²βˆ’πœ†β€²|)(Γ—ξ‚€1βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆ+π›Ώβˆ’πœ†|)π‘Ž(βŸ¨π‘§,π‘§βŸ©),(4.19)𝛿=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²β‰€π›ΎβŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ ξ“πœ…ξ“πœ†,πœ†β€²π›½!(π›½βˆ’πœ†)!𝛽!ξ€·π›½βˆ’πœ†ξ…žξ€Έ!ξ“πœˆ,πœˆβ€²πœŒβŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†ξ…žπœŒβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ Γ—βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ (π›½βˆ’πœŒ)!πœŒβˆ’πœ†π›½βˆ’πœˆβˆ’πœ†πœŒβˆ’πœ†β€²π›½βˆ’πœˆβ€²βˆ’πœ†β€²π›Ώβ€²πœ†β€²π›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώβ€²πœ…!π‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒπ‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒΓ—πΉ(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜈+πœˆβ€²βˆ’π›½+𝜌|)Γ—ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆβ€²+π›Ώβ€²βˆ’πœ†β€²|)ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆ+π›Ώβˆ’πœ†|)π‘Ž(βŸ¨π‘§,π‘§βŸ©),(4.20)𝛿=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²β‰€π›ΎβŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ ξ“πœ…βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώξ…žπœ…βŽžβŽŸβŽŸβŽ ξ“πœ…!𝜈,πœˆβ€²,πœ†,πœ†β€²,πœŒπ›½!(π›½βˆ’πœ†)!𝛽!ξ€·π›½βˆ’πœ†ξ…žξ€Έ!βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†ξ…žπœŒβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ Γ—βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώ(π›½βˆ’πœŒ)!ξ…žπœ†ξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœŒβˆ’πœ†π›½βˆ’πœˆβˆ’πœ†πœŒβˆ’πœ†ξ…žπ›½βˆ’πœˆξ…žβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ π‘§π›Ύβˆ’πœ†βˆ’πœ†β€²βˆ’πœ…+πœŒπ‘§π›Ύβˆ’πœ†βˆ’πœ†β€²βˆ’πœ…+πœŒΓ—πΉ(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜈+πœˆβ€²βˆ’π›½+𝜌|)Γ—ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆβ€²+π›Ώβ€²βˆ’πœ†β€²|)ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆ+π›Ώβˆ’πœ†|)(βŸ¨π‘§,π‘§βŸ©).(4.21) Setting 𝜎=π›½βˆ’πœˆβˆ’πœ† and πœŽβ€²=π›½βˆ’πœˆβ€²βˆ’πœ†β€², we haveπ‘Žπ›Ώ=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²β‰€π›ΎβŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ ξ“πœ…βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώξ…žπœ…βŽžβŽŸβŽŸβŽ ξ“πœ…!πœ†β‰€π›Ώ,πœ†β€²β‰€π›Ώβ€²π›½!(π›½βˆ’πœ†)!𝛽!ξ€·π›½βˆ’πœ†ξ…žξ€Έ!βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώξ…žπœ†ξ…žβŽžβŽŸβŽŸβŽ Γ—ξ“πœ†β‰€πœŒ,πœ†β€²β‰€πœŒ(βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βˆ’πœŒ)!π›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†ξ…žπœŒβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ ξ“πœŽπœŽβ‰€πœŒβˆ’πœ†,β€²β‰€πœŒβˆ’πœ†β€²βŽ›βŽœβŽœβŽπœŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœŒβˆ’πœ†πœŒβˆ’πœ†ξ…žπœŽξ…žβŽžβŽŸβŽŸβŽ Γ—πΉ(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝛽+πœŒβˆ’πœ†βˆ’πœ†β€²βˆ’πœŽβˆ’πœŽβ€²|)Γ—ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜎+𝛿|)ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|πœŽβ€²+𝛿′|)(βŸ¨π‘§,π‘§βŸ©)π‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒπ‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+𝜌,π‘Ž(4.22)𝛿=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ώξ…žβ‰€π›ΎβŽ›βŽœβŽœβŽπ›ΎβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²πœ…βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ ξ“π›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώβ€²πœ…!πœ†β‰€π›Ώ,πœ†ξ…žβ‰€π›Ώξ…žπ›½!(π›½βˆ’πœ†)!𝛽!ξ€·π›½βˆ’πœ†ξ…žξ€Έ!βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ Γ—ξ“π›Ώβ€²πœ†β€²πœ†,πœ†β€²β‰€πœŒβ‰€π›½βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽ(π›½βˆ’πœŒ)!π›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†ξ…žπœŒβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ π‘§π›Ύβˆ’πœ†βˆ’πœ†β€²βˆ’πœ…+πœŒπ‘§π›Ύβˆ’πœ†βˆ’πœ†β€²βˆ’πœ…+πœŒπ‘†πœŒ(βŸ¨π‘§,π‘§βŸ©),(4.23) whereπ‘†πœŒ=ξ“πœŽβ‰€πœŒβˆ’πœ†πœŽξ…žβ‰€πœŒβˆ’πœ†ξ…žβŽ›βŽœβŽœβŽπœŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πΉπœŒβˆ’πœ†πœŒβˆ’πœ†β€²πœŽβ€²(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)𝐹(|𝛽+πœŒβˆ’πœ†βˆ’πœ†β€²βˆ’πœŽβˆ’πœŽβ€²|)ξ‚€1𝐹(|𝜎+𝛿|)ξ‚€1𝐹(|πœŽβ€²+𝛿′|).(4.24) Setting Μ†πœ=(𝜏1,…,πœπ‘›βˆ’1) for a multiindex 𝜏=(𝜏1,…,πœπ‘›), we see thatπ‘†πœŒ=𝐹(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)ξ“Μ†πœ†Μ†Μ†Μ†πœŽβ‰€Μ†πœŒβˆ’πœŽξ…žβ‰€Μ†πœŒβˆ’πœ†ξ…žβŽ›βŽœβŽœβŽΜ†πœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽΜ†Μ†βŽžβŽŸβŽŸβŽ ξ“Μ†πœŒβˆ’Μ†πœŽΜ†πœŒβˆ’πœ†β€²πœŽβ€²πœŽβ€²π‘›β‰€πœŒπ‘›βˆ’πœ†β€²π‘›βŽ›βŽœβŽœβŽπœŒπ‘›βˆ’πœ†ξ…žπ‘›πœŽξ…žπ‘›βŽžβŽŸβŽŸβŽ ξ‚€1𝐹(|πœŽβ€²+𝛿′|)Γ—ξ“πœŽπ‘›β‰€πœŒπ‘›βˆ’πœ†π‘›βŽ›βŽœβŽœβŽπœŒπ‘›βˆ’πœ†π‘›πœŽπ‘›βŽžβŽŸβŽŸβŽ ξ‚΅ξ‚€1𝐹(|𝛿|+𝜎1+β‹―+πœŽπ‘›βˆ’1)ξ‚Ά(πœŽπ‘›)ξ‚€(𝐹)(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²Μ†|+|Μ†πœŒβˆ’πœ†βˆ’Μ†πœŽ|)(πœŒπ‘›βˆ’πœ†π‘›βˆ’πœŽπ‘›).(4.25) Using the Leibnitz ruleπ‘†πœŒ=𝐹(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)ξ“Μ†πœ†Μ†Μ†Μ†πœŽβ‰€Μ†πœŒβˆ’πœŽξ…žβ‰€Μ†πœŒβˆ’πœ†ξ…žβŽ›βŽœβŽœβŽΜ†πœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽΜ†Μ†βŽžβŽŸβŽŸβŽ ξ“Μ†πœŒβˆ’Μ†πœŽΜ†πœŒβˆ’πœ†β€²πœŽβ€²πœŽβ€²π‘›β‰€πœŒπ‘›βˆ’πœ†β€²π‘›βŽ›βŽœβŽœβŽπœŒπ‘›βˆ’πœ†ξ…žπ‘›πœŽξ…žπ‘›βŽžβŽŸβŽŸβŽ ξ‚€1𝐹(|πœŽβ€²+𝛿′|)Γ—ξ‚Έξ‚€1𝐹(|𝛿|+𝜎1+β‹―+πœŽπ‘›βˆ’1)𝐹(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²Μ†|+|Μ†πœŒβˆ’πœ†βˆ’Μ†πœŽ|)ξ‚Ή(πœŒπ‘›βˆ’πœ†π‘›).(4.26) An induction process shows thatπ‘†πœŒ=𝐹(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)ξ“πœŽξ…žβ‰€πœŒβˆ’πœ†ξ…žβŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ ξ‚€1πœŒβˆ’πœ†β€²πœŽβ€²πΉξ‚(|πœŽβ€²+𝛿′|)𝐹(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²|)ξ‚€1𝐹(|𝛿|)ξ‚Ή(|πœŒβˆ’πœ†|).(4.27) Thereforeπ‘Žπ›Ώ=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ώξ…žβ‰€π›ΎβŽ›βŽœβŽœβŽπ›ΎβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²πœ…βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ ξ“π›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώβ€²πœ…!πœ†β‰€π›Ώ,πœ†ξ…žβ‰€π›Ώξ…žπ›½!(π›½βˆ’πœ†)!𝛽!ξ€·π›½βˆ’πœ†ξ…žξ€Έ!βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ Γ—ξ“π›Ώβ€²πœ†β€²πœ†,πœ†β€²β‰€πœŒβ‰€π›½(βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βˆ’πœŒ)!π›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†ξ…žπœŒβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ πΉ(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)(βŸ¨π‘§,π‘§βŸ©)π‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒπ‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒΓ—ξ“πœŽβ€²β‰€πœŒβˆ’πœ†β€²βŽ›βŽœβŽœβŽπœŒβˆ’πœ†ξ…žπœŽξ…žβŽžβŽŸβŽŸβŽ ξ‚€1𝐹(|πœŽβ€²+𝛿′|)𝐹(βŸ¨π‘§,π‘§βŸ©)(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²|)ξ‚€1𝐹(|𝛿|)ξ‚Ή(|πœŒβˆ’πœ†|)(βŸ¨π‘§,π‘§βŸ©).(4.28) Now let 𝜏=π›Ύβˆ’πœ†β€²βˆ’πœ…. It follows thatπ‘Žπ›Ώ=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“πœŒβ‰€π›½ξ“πœβ‰€π›Ύξ“πœ†,πœ†β€²β‰€πœŒξ“πœ†β€²β‰€π›Ώβ€²β‰€πœ+πœ†β€²βŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ύβˆ’π›Ώπ›Ύβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβˆ’πœπ›Ύβˆ’π›Ώξ…žπ›Ύβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ ξ€·βˆ’πœπ›Ύβˆ’πœ†ξ…žξ€Έ!βˆ’πœπ›½!(π›½βˆ’πœ†)!𝛽!ξ€·π›½βˆ’πœ†ξ…žξ€Έ!Γ—βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώξ…žπœ†ξ…žβŽžβŽŸβŽŸβŽ (βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βˆ’πœŒ)!π›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†ξ…žπœŒβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ πΉ(|π›Ύβˆ’π›Ώβˆ’π›Ώβ€²+πœ†β€²+𝜏|)(ξ“βŸ¨π‘§,π‘§βŸ©)πœŽβ€²β‰€πœŒβˆ’πœ†β€²βŽ›βŽœβŽœβŽπœŒβˆ’πœ†ξ…žπœŽξ…žβŽžβŽŸβŽŸβŽ Γ—ξ‚€1𝐹(|πœŽβ€²+𝛿′|)𝐹(βŸ¨π‘§,π‘§βŸ©)(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²|)ξ‚€1𝐹(|𝛿|)ξ‚Ή(|πœŒβˆ’πœ†|)(βŸ¨π‘§,π‘§βŸ©)π‘§πœπ‘§πœπ‘§πœŒβˆ’πœ†π‘§πœŒβˆ’πœ†.(4.29) On the other handβŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ύβˆ’π›Ώξ…žπ›Ύβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ ξ€·βˆ’πœπ›Ύβˆ’πœ†ξ…žξ€Έ!βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ =βˆ’πœπ›Ώβ€²πœ†β€²π›Ύ!ξ€·βˆ’π›Ώξ…ž+πœ†ξ…žξ€Έ+𝜏!πœ†ξ…ž!ξ€·π›Ώξ…žβˆ’πœ†ξ…žξ€Έ!=𝛾!βŽ›βŽœβŽœβŽπœβŽžβŽŸβŽŸβŽ .𝜏!πœ†β€²!π›Ώβ€²βˆ’πœ†β€²(4.30) Thus, if we set 𝜎=π›Ώβ€²βˆ’πœ†β€², thenπ‘Žπ›Ώ=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“πœŒβ‰€π›½ξ“πœβ‰€π›Ύξ“πœ†,πœ†β€²πœŽβ‰€πœŒβ€²β‰€πœŒβˆ’πœ†β€²ξ“πœŽβ‰€πœβŽ›βŽœβŽœβŽπœŒβˆ’πœ†ξ…žπœŽξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ (π›½βˆ’πœŒ)!π›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†β€²πœŒβˆ’πœ†β€²π›Ύ!π‘§πœ+πœŒβˆ’πœ†π‘§πœ+πœŒβˆ’πœ†Γ—βŽ›βŽœβŽœβŽπ›Ύβˆ’π›Ώπ›Ύβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ 1βˆ’πœπœ!πœ†ξ…ž!βŽ›βŽœβŽœβŽπœπœŽβŽžβŽŸβŽŸβŽ πΉ(|π›Ύβˆ’π›Ώ+πœβˆ’πœŽ|)Γ—ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜎+πœŽβ€²+πœ†β€²|)(ξ‚ΈπΉβŸ¨π‘§,π‘§βŸ©)(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²|)ξ‚€1𝐹(|𝛿|)ξ‚Ή(|πœŒβˆ’πœ†|)(βŸ¨π‘§,π‘§βŸ©).(4.31) Using the identityξ“πœŽβ‰€πœβŽ›βŽœβŽœβŽπœπœŽβŽžβŽŸβŽŸβŽ ξ€·πΉ(|π›Ύβˆ’π›Ώ|)ξ€Έ(|πœβˆ’πœŽ|)ξ‚΅ξ‚€1𝐹(|πœŽβ€²+πœ†β€²|)ξ‚Ά(|𝜎|)=ξ‚Έξ‚€1𝐹(|π›Ύβˆ’π›Ώ|)𝐹(|πœŽβ€²+πœ†β€²|)ξ‚Ή(|𝜏|),(4.32) we see thatπ‘Žπ›Ώ=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ύ!πœŒβ‰€π›½πœβ‰€π›Ύξ“πœ†,πœ†β€²πœŽβ‰€πœŒβ€²β‰€πœŒβˆ’πœ†β€²βŽ›βŽœβŽœβŽπœŒβˆ’πœ†ξ…žπœŽξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ 1(π›½βˆ’πœŒ)!π›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†β€²πœŒβˆ’πœ†β€²π›Ύβˆ’π›Ώπ›Ύβˆ’πœ†β€²βˆ’πœπœ!πœ†ξ…ž!π‘§πœ+πœŒβˆ’πœ†π‘§πœ+πœŒβˆ’πœ†Γ—ξ‚ΈπΉ(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²|)ξ‚€1𝐹(|𝛿|)ξ‚Ή(|πœŒβˆ’πœ†|)ξ‚Έξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›Ύβˆ’π›Ώ|)𝐹(|πœŽβ€²+πœ†β€²|)ξ‚Ή(|𝜏|)(βŸ¨π‘§,π‘§βŸ©).(4.33) Notice that, for 𝑝 and π‘ž integers, we have, when 𝑑 is real and 𝑑→+∞,𝐹(𝑝)(𝑑)𝐹(𝑑)βˆΌπ‘‘((1/𝛼)βˆ’1)𝑝,ξ‚€1𝐹(π‘ž)βˆΌπ‘‘(𝑑)((1/𝛼)βˆ’1)𝑝.𝐹(𝑑)(4.34) Hence by Lemmas 4.1 and 4.2 we have the following estimates:||||𝐹(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²|)ξ‚€1𝐹(|𝛿|)ξ‚Ή(|πœŒβˆ’πœ†|)||||(𝑑)≀𝐢𝑑((1/𝛼)βˆ’1)(|𝛽+π›Ώβˆ’πœ†β€²βˆ’πœŽβ€²|)βˆ’|πœŒβˆ’πœ†|,||||ξ‚Έξ‚€1𝐹(|π›Ύβˆ’π›Ώ|)𝐹(|πœŽβ€²+πœ†β€²|)ξ‚Ή(|𝜏|)||||(𝑑)≀𝐢𝑑((1/𝛼)βˆ’1)(|π›Ύβˆ’π›Ώ+πœŽβ€²+πœ†β€²|)βˆ’|𝜏|(4.35) for some constant 𝐢 when the real 𝑑 tends to +∞. Thus for any multiindex 𝛿≀𝛾, there exists a constant 𝐢′ such that π‘Žπ›Ώβ‰€πΆξ…ž|𝑧|2((1/𝛼)βˆ’1)(|𝛽+𝛾|). Then taking the sum over 𝛿 we getξ€·trπ‘†π‘†βˆ—ξ€Έβ‰€πΆξ…žξ…ž|𝑧|2((1/𝛼)βˆ’1)(|𝛽+𝛾|).(4.36) Since Tr(π‘†π‘†βˆ—)=Tr(π‘†βˆ—π‘†) and the operator π‘†βˆ—π‘† is positive, its eigenvalues are bounded above by πΆξ…žξ…ž|𝑧|2((1/𝛼)βˆ’1)(|𝛽+𝛾|). Thus the eigenvalues of (π‘†π‘†βˆ—)1/2 are bounded above by πΆξ…žξ…žξ…ž|𝑧|((1/𝛼)βˆ’1)(|𝛽+𝛾|). Since (π‘†βˆ—π‘†)1/2 is of finite rank, the proof is complete.

Acknowledgments

The author wishes to thank El Hassan Youssfi and Miroslav Englis for useful discussions.